It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, \emph{i.e.} allowing initialized ancilla bits in the input and ignoring ``garbage'' ancilla bits in the output, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate sets, including any gate set that includes the Toffoli or Fredkin gate.
It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, \emph{i.e.} allowing initialized ancilla bits in the input and ignoring ``garbage'' ancilla bits in the output, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate sets, including any gate set that includes the Toffoli or Fredkin gate.
We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical logic from the 1940s. It is a step toward the ambitious goal of classifying all possible quantum gate sets acting on qubits. Our theorem implies a linear-time algorithm (which we have implemented), that takes as input the truth tables of reversible gates G and H, and that decides whether G generates H. Previously, this problem was not even known to be decidable. The theorem also implies that any n-bit reversible circuit can be "compressed" to an equivalent circuit, over the same gates, that uses at most 2^n*poly(n) gates and O(1) ancilla bits; these are the first upper bounds on these quantities known, and are close to optimal. Finally, the theorem implies that every non-degenerate reversible gate can implement either every reversible transformation, or every affine transformation, when restricted to an "encoded subspace." Briefly, the theorem says that every set of reversible gates generates either all reversible transformations on n-bit strings (as the Toffoli gate does); no transformations; all transformations that preserve Hamming weight (as the Fredkin gate does); all transformations that preserve Hamming weight mod k for some k; all affine transformations (as the Controlled-NOT gate does); all affine transformations that preserve Hamming weight mod 2 or mod 4, inner products mod 2, or a combination thereof; or a previous class augmented by a NOT or NOTNOT gate. Ruling out the possibility of additional classes, not in the list, requires some arguments about polynomials, lattices, and Diophantine equations.
Reversible logic has attracted much research interest over the last few decades, especially due to its application in quantum computing. In the construction of reversible gates from basic gates, ancilla bits are commonly used to remove restrictions on the type of gates that a certain set of basic gates generates. With unlimited ancilla bits, many gates (such as Toffoli and Fredkin) become universal reversible gates. However, ancilla bits can be expensive to implement, thus making the problem of minimizing necessary ancilla bits a practical topic. This thesis explores the problem of reversible logic synthesis using a single base gate and a few ancilla bits. Two base gates are discussed: a variation of the 3-bit Toffoli gate and the original 3-bit Fredkin gate. There are three main results associated with these two gates: i) the variated Toffoli gate can generate all n-bit reversible gates using 1 ancilla bit, ii) the variated Toffoli can generate all n-bit reversible gates that are even permutations using no ancilla bit, iii) the Fredkin gate can generate all n-bit conservative reversible gates using 1 ancilla bit. Prior to this paper, the best known result for general universality requires three basic gates, and the best known result for conservative universality needs 5 ancilla bits. The second result is trivially optimal. For the first and the third result, we explicitly prove their optimality: the variated Toffoli cannot generate all n-bit reversible gates without using any extra input lines, and the Fredkin gate cannot generate all n-bit conservative reversible gates without using extra input lines. We also explore a stronger version of the second converse by introducing a new concept called borrowed bits, and prove that the Fredkin gate cannot generate all n-bit conservative reversible gates without ancilla bits, even with an unlimited number of borrowed bits.
Transformation rules play a central role in reversible circuit optimization, template-based rewriting, and equivalence checking, and establishing their completeness is a fundamental problem in reversible logic synthesis. In this work, we investigate the completeness of transformation rules for reversible circuits both with and without ancillary bits and garbage outputs. For reversible circuits without ancillary bits and garbage outputs, we introduce a refined and complete transformation rule set RCr, obtained by replacing one rule in the rule set RC proposed in the previous work (TCAD, 45, pp. 3711--3724, 2026) with a simpler and widely adopted transformation rule. Since all rules in RCr are commonly used in reversible logic synthesis, this result reveals that practically adopted transformation rules are already sufficient to establish a complete rewriting framework. Based on this result, we further propose an extended rule set, denoted by RC+, and prove its completeness for reversible circuits with ancillary bits and garbage outputs. To the best of our knowledge, this work presents the first complete transformation rule set for arbitrary reversible circuits, regardless of whether ancillary bits or garbage outputs are employed. The proposed framework establishes a theoretical foundation for circuit optimization, template generation, and equivalence checking, and may facilitate the development of automated design tools for reversible and quantum circuits.
Reversible logic has applications in quantum computing, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate and the Fredkin gate. Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply transformations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified-they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously.
Reversible logic is expanding grounds with its applicability to various upcoming technologies and quantum computation. Numerous reversible gates has also been proposed for designing logic circuits apart of basic quantum controlled Toffoli and Fredkin gates. A comparative study and analysis of a range of reversible gates from the literature is presented in this paper to obtain an optimal solution for designing these circuits. The gates are synthesized and implemented to obtain equivalent quantum circuits and operating cost in terms of number of inputs, gate cost, quantum cost, garbage output and ancilla input. The comparison is done at two subsequent levels i.e. gates structural level and circuit design level, providing a better solution for designing logic circuits. It is calculated that, basic gates are 50% efficient at gate level and 33% at design level analysis in terms of gate cost, as compared to the most efficient gates proposed in the literature.
Abstract—Reversible logic has applications in quantum com-puting, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate (which performs a controlled NOT) and the Fredkin gate (performs a controlled swap of two inputs). Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply trans-formations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified—they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously. I.